Complete root distribution with respect to parabolas and some results with respect to hyperbolas and sectors

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Abstract
Criteria are derived to determine the distribution of the characteristic roots of a Byatem with respect to parabolas and hyperbolas in the complex plane, which are related to certain physical properties of the system. The criteria are given in terms of a classical half-plane algorithm, e.g. a Routh array, carried out on a transformed polynomial of double the degree of the original characteristic polynomial. For parabolas, left, right or shifted, the algorithm provides the complete root distribution including criteria for root clustering and root exclusion as Bpecial cases. For hyperbolas, the algorithm provides in all cases criteria for root clustering inside the (open or closed) left or right branch of the hyperbola, and for characteristic polynomials which are Hurwitz, the algorithm provides the complete root distribution with respect to the left branch of the hyperbola. Also, by an additional classical test (Sturm), the algorithm provides in all cases the exact number of roots on the left boundary of the hyperbola and on the right boundary of the hyperbola. By a limiting process the results on hyperbolas are extended to linear sectors, and by another limiting process the results are extended to symmetrical intervals on the real axis.

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